> I omitted subscripts for α and 1: I assume this is fine because they can never be ambiguous. Am I right in this assumption?

We write subscripts for \$$\alpha\$$ since \$$\alpha\$$ is really a natural transformation which transform one functor (doing the monoidal product twice on the left) to another functor (doing the monoidal product twice on the right). The subscripts give a component of the associator (a morphism that switches the monoidal product of a particular set of three objects).

Technically, you need the subscripts to get a morphism from the associator and then compose that morphism with other morphisms. There's a similar story for the identity morphism (although we also use 1 for the identity functor and identity natural transformation (which is just the identity morphism on each object)). It's not that it's never ambiguous, but usually context makes it clear. And that's kinda the whole point of string diagrams. In the string diagrams, these subtle technical distinctions don't end up mattering.